Welby, who has only been a bishop for a year, said he didn't expect to be chosen and described himself as "overwhelmed and surprised," The Daily Telegraph reported.
News of the appointment was leaked earlier this week and was formally announced by Prime Minister David Cameron at Lambeth Palace in London.
The new archbishop has two hot-button issues to deal with. Welby said he supports consecrating women as bishops and suggested his views on gay marriage are shifting toward supporting same-sex unions. He promised to "listen to the voice of the LGBT communities and examine my own thinking."
"We must have no truck with any form of homophobia in any part of the church," he said.
When he is installed, Welby will also become head of the Anglican Communion, a fractious coalition of national and regional churches embracing 77 million Anglicans worldwide.
Welby, 56, is a former oil executive who joined the clergy in 1992 after feeling "a sense of God calling," The New York Times reported.
He will succeed the Most Rev. Rowan Williams, who announced in March that he would step down.
Welby will be the 105th archbishop of Canterbury, with predecessors who include St. Thomas Becket and Thomas Cranmer, who led the church through its separation from the Roman Catholic Church and was burned at the stake in 1556 after Mary I, a Catholic, became queen.
Like them, Welby heads a church divided, now because of disagreements over the role of homosexuals and women in the church.
He had worked for a decade in the oil industry in Europe when his daughter, Johanna, the youngest of his six children, was killed in a car crash in 1983. Welby said while he and wife, Caroline, struggled to deal with Johanna's death "it actually brought us closer to God."
He left his executive position and went into training for the priesthood in 1987 because "I was unable to get away from a sense of God calling."
Welby's new role must still be formally approved by Queen Elizabeth II and endorsed by the college of canons at Canterbury Cathedral.
Welby, who became the bishop of Durham in November 2011, is expected to be installed March 21.